Central limit theorem for linear processes with values in Hilbert space (Q1382473)

The author studies the behaviour of the sums of i.i.d. sequences of Hilbert space valued random vectors of the form \(S_n=\sum _{k=1}^n a_{kn}\varepsilon _k,\) where \(\varepsilon _k\) are i.i.d. \(H\)-valued random vectors with \(E(\varepsilon _k)=0\) and \(E|\varepsilon _k|^2=\sigma ^2_\varepsilon <\infty \) and the components of the triangular array \((a_{nk})\) are bounded linear operators on Hilbert space \(H\). Under the assumptions \( \sup _n \sum _{k=1}^n |a_{kn}|^2 <\infty\), \(\max _{1\leq k\leq n} |a_{kn}|\rightarrow 0\) if \(n\to \infty \) it is proved that \(S_n\) converges in distribution to a centered \(H\)-valued Gaussian random vector with the covariance operator \(T=(\sigma _{ij})\) if and only if for some orthonormal basis \((e_i)\) in \(H\) \[ \lim _{n\to \infty }E\langle S_n,e_i\rangle \langle S_n,e_j \rangle = \sigma _{ij},\quad \limsup _{n\to \infty } E|S_n|^2 = \sum _i \sigma _{ii}. \]